<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.2 beta6 (August 14th, 1998)
original version by:  Nikos Drakos, CBLU, University of Leeds
* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Further Details: Error Bounds for Linear Least Squares
Problems</TITLE>
<META NAME="description" CONTENT="Further Details: Error Bounds for Linear Least Squares
Problems">
<META NAME="keywords" CONTENT="lug_l2h">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="lug_l2h.css">
<LINK REL="previous" HREF="node82.html">
<LINK REL="up" HREF="node82.html">
<LINK REL="next" HREF="node84.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html5353"
 HREF="node84.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next_motif.gif"></A> 
<A NAME="tex2html5347"
 HREF="node82.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up_motif.gif"></A> 
<A NAME="tex2html5343"
 HREF="node82.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="previous_motif.gif"></A> 
<A NAME="tex2html5349"
 HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents_motif.gif"></A> 
<A NAME="tex2html5351"
 HREF="node152.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index_motif.gif"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html5354"
 HREF="node84.html">Error Bounds for Generalized</A>
<B> Up:</B> <A NAME="tex2html5348"
 HREF="node82.html">Error Bounds for Linear</A>
<B> Previous:</B> <A NAME="tex2html5344"
 HREF="node82.html">Error Bounds for Linear</A>
 &nbsp <B>  <A NAME="tex2html5350"
 HREF="node1.html">Contents</A></B> 
 &nbsp <B>  <A NAME="tex2html5352"
 HREF="node152.html">Index</A></B> 
<BR>
<BR>
<!--End of Navigation Panel-->

<H2><A NAME="SECTION03451000000000000000"></A><A NAME="secbackgroundlsq"></A>
<BR>
Further Details:  Error Bounds for Linear Least Squares
Problems
</H2>

<P>
The conventional error analysis of linear least squares problems goes
as follows<A NAME="10832"></A>.
As above, let <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.gif"
 ALT="$\hat{x}$">
be the solution to minimizing
<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img437.gif"
 ALT="$\Vert Ax-b \Vert _2$">
computed by
LAPACK using one of the least squares drivers
xGELS, xGELSX, xGELSY, xGELSS or xGELSD
(see subsection <A HREF="node27.html#subsecdrivellsq">2.3.2</A>).
We discuss the most common case, where <B><I>A</I></B> is
overdetermined<A NAME="10834"></A>
(i.e., has more rows than columns) and has full rank
[<A
 HREF="node151.html#bjorck3">16</A>,<A
 HREF="node151.html#demmelMA221">25</A>,<A
 HREF="node151.html#GVL2">55</A>,<A
 HREF="node151.html#higham96">67</A>]:
<A NAME="10836"></A><A NAME="10837"></A><A NAME="10838"></A><A NAME="10839"></A>
<A NAME="10840"></A><A NAME="10841"></A><A NAME="10842"></A><A NAME="10843"></A>
<A NAME="10844"></A><A NAME="10845"></A><A NAME="10846"></A><A NAME="10847"></A>
<A NAME="10848"></A><A NAME="10849"></A><A NAME="10850"></A><A NAME="10851"></A>
<A NAME="10852"></A><A NAME="10853"></A><A NAME="10854"></A><A NAME="10855"></A>

<P>
<BLOCKQUOTE>
The computed solution <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.gif"
 ALT="$\hat{x}$">
has a small normwise backward error.
In other words <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.gif"
 ALT="$\hat{x}$">
minimizes 
<!-- MATH
 $\|(A+E) \hat{x}- (b+f)\|_2$
 -->
<IMG
 WIDTH="175" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img449.gif"
 ALT="$\Vert(A+E) \hat{x}- (b+f)\Vert _2$">,
where
<B><I>E</I></B> and <B><I>f</I></B> satisfy
<A NAME="10857"></A>
<A NAME="10858"></A>
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\max \left( \frac{\| E \|_2}{\| A \|_2} ,
            \frac{\| f \|_2}{\| b \|_2} \right) \leq p(n) \epsilon
\end{displaymath}
 -->


<IMG
 WIDTH="211" HEIGHT="48" BORDER="0"
 SRC="img450.gif"
 ALT="\begin{displaymath}
\max \left( \frac{\Vert E \Vert _2}{\Vert A \Vert _2} ,
\frac{\Vert f \Vert _2}{\Vert b \Vert _2} \right) \leq p(n) \epsilon
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
and <B><I>p</I>(<I>n</I>)</B> is a modestly growing function of <B><I>n</I></B>. We take <B><I>p</I>(<I>n</I>)=1</B> in
the code fragments above.
Let 
<!-- MATH
 $\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$
 -->
<IMG
 WIDTH="203" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img451.gif"
 ALT="$\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$">
(approximated by
1/<TT>RCOND</TT> in the above code fragments),

<!-- MATH
 $\rho = \|A \hat{x} -b\|_2$
 -->
<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img452.gif"
 ALT="$\rho = \Vert A \hat{x} -b\Vert _2$">
(= <TT>RNORM</TT> above), and 
<!-- MATH
 $\sin ( \theta ) = \rho / \|b\|_2$
 -->
<IMG
 WIDTH="122" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img453.gif"
 ALT="$\sin ( \theta ) = \rho / \Vert b\Vert _2$">
(<TT>SINT = RNORM / BNORM</TT> above). Here, <IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img328.gif"
 ALT="$\theta$">
is the acute angle between
the vectors <IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img454.gif"
 ALT="$A \hat{x}$">
and <B><I>b</I></B>.
<A NAME="10870"></A>
<A NAME="10871"></A>
Then when <IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img455.gif"
 ALT="$p(n) \epsilon$">
is small, the error <IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img456.gif"
 ALT="$\hat{x}- x$">
is bounded by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\frac{\|x-\hat{x}\|_2}{\|x\|_2} \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}p(n) \epsilon
\left\{ \frac{2 \kappa_2 (A)}{\cos ( \theta )} + \tan ( \theta ) \kappa_2^2 (A)
\right\},
\end{displaymath}
 -->


<IMG
 WIDTH="334" HEIGHT="48" BORDER="0"
 SRC="img457.gif"
 ALT="\begin{displaymath}
\frac{\Vert x-\hat{x}\Vert _2}{\Vert x\Vert _2} \mathrel{\ra...
...)}{\cos ( \theta )} + \tan ( \theta ) \kappa_2^2 (A)
\right\},
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
where 
<!-- MATH
 $\cos ( \theta )$
 -->
<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img458.gif"
 ALT="$\cos ( \theta ) $">
= <TT>COST</TT> and 
<!-- MATH
 $\tan ( \theta )$
 -->
<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img459.gif"
 ALT="$\tan ( \theta )$">
= <TT>TANT</TT> in the code fragments
above.

</BLOCKQUOTE>

<P>
We avoid overflow by making sure <TT>RCOND</TT> and <TT>COST</TT> are both at least
<IMG
 WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img420.gif"
 ALT="$\epsilon =$">
<TT>EPSMCH</TT>, and by handling the case of a zero <TT>B</TT> matrix
separately (<TT>BNORM = 0</TT>).
<A NAME="10884"></A>
<A NAME="10885"></A>

<P>

<!-- MATH
 $\kappa_2 (A) = \sigma_{\max} (A) / \sigma_{\min} (A)$
 -->
<IMG
 WIDTH="203" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img451.gif"
 ALT="$\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$">
may be computed directly
from the singular values of <B><I>A</I></B> returned by xGELSS or xGELSD (as in the code fragment)
or by xGESVD or xGESDD. It may also be approximated by using xTRCON following calls to
xGELS, xGELSX or xGELSY.  xTRCON estimates 
<!-- MATH
 $\kappa_{\infty}$
 -->
<IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img460.gif"
 ALT="$\kappa_{\infty}$">
or <IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img461.gif"
 ALT="$\kappa_1$">
instead
of <IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img462.gif"
 ALT="$\kappa_2$">,
but these can differ from <IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img462.gif"
 ALT="$\kappa_2$">
by at most a factor of <B><I>n</I></B>.
<A NAME="10889"></A><A NAME="10890"></A><A NAME="10891"></A><A NAME="10892"></A>
<A NAME="10893"></A><A NAME="10894"></A><A NAME="10895"></A><A NAME="10896"></A>
<A NAME="10897"></A><A NAME="10898"></A><A NAME="10899"></A><A NAME="10900"></A>

<P>
If <B><I>A</I></B> is rank-deficient, xGELSS, xGELSD, xGELSY and xGELSX can be used to
<B>regularize</B> the
problem<A NAME="10902"></A><A NAME="10903"></A>
by treating all singular values
less than a user-specified threshold
(
<!-- MATH
 ${\tt RCND} \cdot \sigma_{\max} (A)$
 -->
<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img463.gif"
 ALT="${\tt RCND} \cdot \sigma_{\max} (A)$">)
as
exactly zero.  The number of singular values treated as nonzero is returned
in <TT>RANK</TT>.  See [<A
 HREF="node151.html#bjorck3">16</A>,<A
 HREF="node151.html#GVL2">55</A>,<A
 HREF="node151.html#higham96">67</A>]
for error bounds in this case, as well as
<A NAME="10908"></A>
[<A
 HREF="node151.html#demmelhigham1">28</A>] for the
underdetermined<A NAME="10910"></A>
<A NAME="10911"></A> case.
The ability to deal with rank-deficient matrices is the principal attraction
of these four drivers, which are more expensive than the simple driver xGELS.

<P>
The solution of the overdetermined,
<A NAME="10912"></A>
<A NAME="10913"></A>
full-rank problem may also be
characterized as the solution of the linear system of equations
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\left( \begin{array}{cc} I & A \\A^T & 0 \end{array} \right) \left( \begin{array}{c} r \\x \end{array} \right) =
\left( \begin{array}{c} b \\0 \end{array} \right) .
\end{displaymath}
 -->


<IMG
 WIDTH="225" HEIGHT="54" BORDER="0"
 SRC="img464.gif"
 ALT="\begin{displaymath}
\left( \begin{array}{cc} I &amp; A \\ A^T &amp; 0 \end{array} \right...
...\right) =
\left( \begin{array}{c} b \\ 0 \end{array} \right) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
By solving this linear system using xyyRFS or xyySVX (see section
<A HREF="node80.html#secAx_b">4.4</A>) componentwise error bounds can also be obtained
[<A
 HREF="node151.html#arioliduffderijk">7</A>].

<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html5353"
 HREF="node84.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next_motif.gif"></A> 
<A NAME="tex2html5347"
 HREF="node82.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up_motif.gif"></A> 
<A NAME="tex2html5343"
 HREF="node82.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="previous_motif.gif"></A> 
<A NAME="tex2html5349"
 HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents_motif.gif"></A> 
<A NAME="tex2html5351"
 HREF="node152.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index_motif.gif"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html5354"
 HREF="node84.html">Error Bounds for Generalized</A>
<B> Up:</B> <A NAME="tex2html5348"
 HREF="node82.html">Error Bounds for Linear</A>
<B> Previous:</B> <A NAME="tex2html5344"
 HREF="node82.html">Error Bounds for Linear</A>
 &nbsp <B>  <A NAME="tex2html5350"
 HREF="node1.html">Contents</A></B> 
 &nbsp <B>  <A NAME="tex2html5352"
 HREF="node152.html">Index</A></B> 
<!--End of Navigation Panel-->
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
</BODY>
</HTML>
